The K-functional and saturated linear approximation processes

Abstract It is shown that already the knowledge of saturation class $$\, {{\mathfrak {F}}}(J_t, L^p) $$ F ( J t , L p ) a saturated convolution approximation process \{J_t\}$$ { } on L^p({{\mathbb {R}}}^n),\, 1\le p< \infty ,$$ R n 1 ≤ < ∞ completely determines its norm behavior. This achieved by using K$$ K -functional K(t,f;L^p,{{\mathfrak L^p)) f ; as comparison scale, which relates one hand and other appropriate moduli smoothness. implies simultaneously gets so-called direct inverse theorems. There are open problems if order slightly perturbed, e.g., |\log t|^\lambda | log λ -factor, \lambda >0.$$ > 0 . Proofs mainly based Fourier transformation.